Bounds on the number of automorphisms of curves over algebraically closed fields

ISRAEL JOURNAL OF MATHEMATICS 194 (2013), 69–76

DOI: 10.1007/s11856-012-0078-0

BOUNDS ON THE NUMBER OF AUTOMORPHISMS OFCURVES OVER ALGEBRAICALLY CLOSED FIELDS

BY

Barry Green

African Institute for Mathematical Sciences

Muizenberg, 9745 Cape Town, South Africa

and

Department of Mathematical Sciences, University of Stellenbosch

7602 Stellenbosch, South Africa

e-mail: [email protected]

ABSTRACT

In this note we compare the maximum number of automorphisms of smooth

projective curves of genus g ≥ 2 over algebraically closed fields of charac-

teristic 0 or a prime p > 0, respectively. For given g ≥ 2, we show that for

almost all primes p > 0, this number is the same.

1. Introduction

The aim of this note is give partial answers to questions posed by Oort in [O]

on the group of automorphisms of smooth projective curves over algebraically

closed fields. We investigate the bounds on the possible orders such groups

can have and compare what is known in characteristic zero with questions and

conjectures in positive characteristic, some of which are answered here.

Let K be an algebraically closed field and let C/K be a smooth projective

curve with genus gC , which is assumed greater than or equal to 2 throughout

this note. We are interested in #(AutK(C)) and recall that if char(K) = 0,

then by results of Hurwitz, #(AutK(C)) ≤ 84(gC − 1) and that this is also true

if char(K) = p > 2gC + 1. See for example [Sa], p. 582, or [R2] for a stronger

result with one exception.

Received July 12, 2011

69

70 BARRY GREEN Isr. J. Math.

We are interested in the lower and upper bounds of

μ(g, char(K)) := maxgC=g

#((AutK(C)),

and note that if char(K) = 0 then results of Macbeath, [M], Maclachlan, [Mc],

and Accola, [A1], prove that

8(g + 1) ≤ μ(g, 0) ≤ 84(g − 1)

with each of the bounds being attained for infinitely many values of g. The

result on the upper bound was demonstrated in [M] and for the lower bounds

in [Mc] and [A1].

1.1. Question (Oort) [O]): If p is any prime and K is an algebraically closed

field with char(K) = p, does there exist a non-zero polynomial Mp(T ) ∈ Q[T ]

such that:

(i) For all g ≥ 2, Mp(g) ≤ μ(g, p).

(ii) For infinitely many g, Mp(g) = μ(g, p).

Furthermore, if such a polynomial exists, is it the same for different values of

p and is it equal to 8(g + 1)?

1.2. We prove:

(i) Given g ≥ 2, for almost all primes p > 0, μ(g, p) ≥ 8(g + 1).

(ii) For each positive integer m there exist integers 2 ≤ g1 < g2 < · · · < gm

and 2 ≤ g′1 < g′2 < · · · < g′m, such that for 1 ≤ i ≤ m :

μ(gi, p) = μ(gi, 0) = 8(gi + 1) for almost all p > 84(gi − 1) and

μ(g′i, p) = μ(g′i, 0) = 84(g′i − 1) for all p > 84(g′i − 1).

For a fixed prime p these results don’t answer Oort’s question, but we

will prove a weaker result, namely:

(iii) Let g ≥ 2 with μ(g, 0) = 8(g + 1). Then for almost all primes p > 0,

μ(g, p) = 8(g + 1).

Acknowledgement. I would like to thank Moshe Jarden for helpful sugges-

tions for improving this note.

Vol. 194, 2013 BOUNDS ON THE AUTOMORPHISM GROUP OF CURVES 71

2. Good reduction of curves and their automorphism groups

Let K be an algebraically closed field of characteristic zero and M be the set of

non-archimedean valuations on K of mixed characteristic. For each v ∈ M let

Kv denote the residue field, which has positive characteristic, p say. Let C/K

be a smooth projective irreducible curve of genus gC ≥ 2. Then for almost all

primes p, there exist v ∈ M of residue characteristic p, such that C has good

reduction at v; see for example [D], [R] or [GMP I]. We shall denote the reduced

curve by Cv/Kv, which is smooth, projective and has genus gCv = gC .

In this situation there is a naturally well defined homomorphism

Φ:AutK(C) −→ AutKv(Cv).

For convenience, we recall how this homomorphism is constructed: Let F =

κ(C) denote the function field of C, and recall that the reduction of the curve

C at v corresponds to a valuation prolongation of v on K to F (called a con-

stant reduction), which we also denote by v. We show that the action of the

automorphisms of F |K on this valuation v is invariant. If τ ∈ Aut(F |K), then

w = v ◦ τ−1 is a valuation of F |K whose valuation ring and maximal ideals are

τ(Ov) and τ(Mv), respectively. Let Fv, respectively Fw, denote the residue

fields of F with respect to v, respectively w. It follows directly that the map

τ : Fv → Fw := τ(Ov)/τ(Mv)

defined by fv �→ τ(f) + τ(Mv) is a well-defined isomorphism that fixes each

element of Kv. Hence the function fields Fv|Kv and Fw|Kv have the same

genus. We deduce that w = v ◦ τ−1 is also a good reduction of F |K and by

the uniqueness of the good reduction over a given valuation of K (see [GMP I],

Corollary 3.2), it follows that w = v, so Fw = Fv. Hence we have a well defined

homomorphism

Ψ:Aut(F |K) −→ Aut(Fv|Kv).

Observing that κ(Cv) = Fv this map induces the homomorphism Φ above.

Theorem 2.1 ([K]): Let Φ : AutK(C) −→ AutKv(Cv) be as defined above.

Then:

(a) Φ is injective.


(b) For each G ⊂ AutK(C), the restriction ΦG : G −→ AutKv(Cv/Dv) is

surjective, where D := C/G.1

The proofs of these results can be found in [K] in the context of function

fields and their constant reductions. As we shall be using (a) in what follows

we include a short proof here for our context and notation.

Proof of (a). Suppose σ ∈ KerΦ with σ = 1C , and by abuse of notation we

denote its action on the function field F = κ(C) by σ too. We first observe

that σ ∈ ker(Φ) if and only if σ(x) − x ∈ Mv for each x ∈ Ov ⊂ F. Therefore,

ker(Φ) is the inertia group, say T, of the the extension F |E, where E = FG

and G = Aut(F |K) ≈ AutK(C). Now as F |K has good reduction at v it

satisfies, v(F×) = v(K×), so v(F×) = v(E×), [GMP I], Corollary 3.2. By [ZS],

Chapter VI, §12, there is a homomorphism of the inertia group T of F |E into

Hom(v(F×)/v(E×), (Fv)×) whose kernel is the ramification group V. It follows

that T = V . It is well known (see for example [ZS], Theorem 24, p. 77) that this

group is a p-Sylow subgroup and Fv|Ev is a purely inseparable extension. We

have shown that KerΦ is a p-group. (See also [E], Chapter III, or [Ef], Chapter

16, for these facts.)

We next examine the covering C → D := C/〈σ〉 and its reduction with

respect to v, Cv → Dv. Then as 〈σ〉 is a p-group both coverings determine p

extensions of the function fields, with the latter being purely inseparable as Kv

has characteristic p. Now since the extension of function fields determined by Cv

and Dv is purely inseparable, it follows that these fields are isomorphic (see for

example [St], Proposition 3.10.2) and so they have the same genus. Therefore

Cv and Dv have the same geometric genus, say g, which is equal to gC as C

has good reduction at v.

Noting that the genus of D is greater than or equal to the geometric genus

of its reduction Dv, we obtain the following inequality of genera:2

gC ≥ gD ≥ g = gCv = gC .

Hence it follows that we have equalities and so D has good reduction at v and

gC = gD. Finally, by the Riemann–Hurwitz formula and since gC ≥ 2, this

1 Although not needed in this note, we remark that as C/K has good reduction at v, so

does D/K. See for example [Y] and [G].2 See for example [GMP I], Corollary 2.5, for a proof of this fact in the context of constant

reductions.


can only happen if C = D and consequently σ = 1C , a contradiction. This

completes the proof of (a).

Question: Under which conditions is Φ surjective?

Theorem 2.2 below gives a partial answer to this question:

Theorem 2.2: Theorem 2.2. Let C/K and Cv/Kv be as above and suppose

the order of G = AutKv(Cv) is relatively prime to p. Then Φ is an isomorphism.

Proof. By Grothendieck (SGA 1, Expose XIII, §2) for C/K as above we can

lift G to G ⊂ AutK(C) such that G ∼= G. Since Φ is injective the result follows.

See also [CGH], sections 1 and 2.

Corollary 2.3: Let C/K be a smooth projective irreducible curve which has

good reduction at v ∈ M. If char(Kv) = p > 84(gC − 1)), then Φ is an isomor-

phism.

Proof. Since C/K has good reduction at v and p > 84(gC − 1), we have

p > 84(gCv − 1) > 2gCv − 1. Hence, by the second paragraph of the intro-

duction it follows that

#(AutKv(Cv)) ≤ 84(gCv − 1).

Therefore (#(AutKv(Cv)), p) = 1 and, by Theorem 2.2, Φ is an

isomorphism.

Corollary 2.4: Let g ≥ 2 be given. Then for all primes p > 84(g − 1),

μ(g, 0) ≥ μ(g, p). If in addition μ(g, p) = 84(g − 1), then μ(g, 0) = μ(g, p).

Proof. Let K be an algebraically closed field of characteristic p and C/K be a

smooth projective irreducible curve with genus g and #(AutK(C)) = μ(g, p).

Next suppose (C/K, v) is a good lifting of C/K to characteristic 0, with K

algebraically closed and Kv = K. Then as p > 84(g − 1), Theorem 2.2 implies

that Φ is an isomorphism and μ(g, 0) ≥ #(AutK(C)) = #(AutK(C)) = μ(g, p).

The final assertion follows directly from the upper bound on the number of

automorphisms in characteristic 0.

We next recall a general fact on the existence of good reductions of smooth

projective irreducible curves C/K. For a proof see [D], [R1], or the observation

[GMP I], 1.3.


2.5. Deuring/Roquette. Let C/K be a smooth projective irreducible curve

and suppose F = κ(C). Then for each non-constant f ∈ F, there exists a finite

set A ⊂ K such that for each v ∈ M, if v(A) ≥ 0, then the Gauss valuation vf

on K(f) has a unique extension to F, which is a good reduction of C/K. Hence

for almost all primes p, there exists v ∈ M of residue characteristic p, such that

C has good reduction at v.

Proposition 2.6: Let g ≥ 2 be given. Then for almost all primes p, μ(g, 0) ≤μ(g, p). If in addition p > 84(g− 1) then for almost all such p, μ(g, 0) = μ(g, p).

Proof. Given g ≥ 2, let C/K be chosen so that gC = g and #(AutK(C)) =

μ(g, 0). Then by 2.5 above, for almost all primes p there exists v defined on K

so that char(Kv) = p and C/K has good reduction at v. It follows by Theorem

2.1 that μ(g, 0) = #(AutK(C)) ≤ #(AutKv(Cv)) ≤ μ(g, p). The final assertion

follows by combining this result with 2.4 above.

We return to Oort’s question:

3. Bounds on the number of automorphisms

We now prove the results stated in 1.2.

Proposition 3.1: Let g ≥ 2 be given. Then for almost all primes p > 0,

μ(g, p) ≥ 8(g + 1).

Proof. By Proposition 2.6 for almost all p > 0, μ(g, p) ≥ μ(g, 0). Therefore

since μ(g, 0) ≥ 8(g + 1) for all g ≥ 2, the result follows.

Applying Theorem 2.1 we prove 1.2 (iii) from the introduction. For conve-

nience we first define S := {g ∈ N | μ(g, 0) = 8(g + 1) }, which is an infinite set

by [Mc], Theorem 5.

Proposition 3.2: Let g ∈ S. Then for almost all primes p > 0, μ(g, p) =

μ(g, 0) = 8(g + 1).

Proof. This follows directly from Proposition 2.6.

Theorem 3.3: For each positive integer m there exist integers

2 ≤ g1 < g2 < · · · ≤ gm and 2 ≤ g′1 < g′2 < · · · < g′m,

such that for 1 ≤ i ≤ m,


(i) μ(gi, p) = μ(gi, 0) = 8(gi + 1) for almost all primes p > 84(gi − 1),

(ii) μ(g′i, p) = μ(g′i, 0) = 84(g′i − 1) for all primes p > 84(g′i − 1).

Proof. (i) For m ∈ N, choose an increasing sequence g1, g2, . . . , gm ∈ S. Then

for each i, 1 ≤ i ≤ m, by 2.6, μ(gi, p) = μ(gi, 0) = 8(gi+1) for almost all primes

p > 84(gi − 1).

(ii) For m ∈ N, choose an increasing sequence

g′1, g′2, . . . , g

′m ∈ T := {g′ ∈ N | μ(g′, 0) = 84(g′ − 1) },

which is infinite and by [M], Corollary to Theorem 6. Then ii) follows immedi-

ately from 2.4 for all primes p > 84(g′i − 1).

References

[A1] R. D. M. Accola, On the number of automorphisms of a closed Riemann surface,

Transactions of the American Mathematical Society 131 (1968), 398–408.

[A2] R. D. M. Accola, Topics in the Theory of Riemann Surface, Lecture Notes in Math-

ematics 1595, Springer-Verlag, Berlin–Heidelberg–New York, 1994.

[CGH] T. Chinburg, R. Guralnick and D. Harbater, Oort groups and lifting problems,

Compositio Mathematica 144 (2008), 849–866.

[D] M. Deuring, Reduktion algebriascher Funktionenkorper nach Primdivisoren des

Konstantenkorpers, Mathematische Zeitschrift 47 (1942), 643–654.

[Ef] I. Efrat, Valuations, Orderings, and Milnor K-theory, Mathematical Surveys and

Monographs, Vol. 124, American Mathematical Society, Providence, RI, 2006.

[E] O. Endler, Valuation Theory, Springer Universitext, Berlin–Heidelberg–New York,

1972.

[GMP I] B. Green, M. Matignon and F. Pop, On valued function fields I, Manuscripta Math-

ematica 65 (1989), 257–276.

[G] B. Green, The relative genus inequality for curves over valuation rings, Journal of

Algebra 181 (1996), 836–856.

[K] H. Knaf, Reduktion von Funktionenkorpern und Automorphismengruppen, Diplo-

marbeit, Universitat Heidelberg, 1990.

[M] A. M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5 (1961),

90–96.

[Mc] C. Maclachlan, A bound for the number of automorphisms of a compact Riemann

surface, Journal of the London Mathematical Society 44 (1961), 265–272.

[O] F. Oort, Minimal maximal number of automorphisms of curves, in Problems

from the Workshop on Automorphisms of Curves, Lorentz Center, pp. 40–41,

arXiv:math/0411059v1 [math.AG], 3 Nov. 2004.

[R1] P. Roquette, Zur Theorie der Konstantenreduktion algebraischer Mannig-

faltigkeiten, Journal fur die Reine und Angewandte Mathematik 200 (1958), 1–44.


[R2] P. Roquette, Abschatzung der Automorphismenzahl von Funktionenkorpern bei

Primzahlcharacteristik, Mathematische Zeitschrift 117 (1970), 157–163.

[Sa] G. D. V. Salvador, Topics in the Theory of Algebraic Function Fields, Mathematics:

Theory & Applications, Birkhauser, Boston, 2006.

[St] H. Stichtenoth, Algebraic Function Fields and Codes, Graduate Texts in Mathemat-

ics 254, Springer-Verlag, Berlin–Heidelberg, 2009.

[Y] T. Youssefi, Inegalite relative des genres, Manuscripta Mathematica 78 (1993), 111–

128.

[ZS] O. Zariski and P. Samuel, Commutative Algebra, Vollume II, Graduate Texts in

Mathematics 29, Springer-Verlag, New York–Heidelberg–Berlin, 1975.

Documents

Bounds on the number of automorphisms of curves over algebraically closed fields